Hi, for the Galois group I'm using a rather naive approach for equations of degree 4 or less - it's based on the theory given in a standard textbook (Artin's Algebra, 2nd edition) - looking at the discriminant and in the quartic case at the resolvent cubic. It's a solid algorithm except for one of the cases where there is ambiguity between C_4 and D_4, but that's OK for most of the polynomials you'd encounter in the exercises... But now that I looked around I saw there might be some more general algorithms out there (http://mathoverflow.net/questions/ 22923/computing-the-galois-group-of-a-polynomial/ for example), so that's probably what I'm going to try next.
Respectively, representing groups will be just by name (not so many of them in degrees less than 5...) but obviously in higher degrees it'd be convenient to have a way of handling arbitrary groups. Galois groups are naturally embedded in the symmetric groups S_n, so permutations will probably pop up from the algorithms; as size gets larger (approx. n!) it'd be better to have them in cycle notation, and also try to extract a list of generators / generators with relations. Other than that, I don't know... I'm currently doing representation theory and it sounds tempting (eigenvals()? character tables?). But that's just the mathematical side of it, and I'll have to think a lot about implementation. Aleksandar Makelov On Mar 16, 4:32 pm, Aaron Meurer <asmeu...@gmail.com> wrote: > As others have said, we will leave the straight graph theory to > networkx and similar libraries. Those other things would be fitting, > though. Take a look at what's already implemented in the > combinatorics module, the sets module, and elsewhere. > > And it would be awesome to have a group theory module. We presently > only have a Permutation class in the combinatorics module, but other > than that, we don't really have a good way to represent a group. > Obviously, to compute the Galois group of a polynomial, you need a way > to represent it, so for this idea, you would really need to implement > a group theory framework that we can build upon. > > What algorithm do you use to compute the Galois group? > > Aaron Meurer > > On Fri, Mar 16, 2012 at 6:06 AM, Aleksandar Makelov > > > > > > > > <amake...@college.harvard.edu> wrote: > > Hi guys, > > > I'm currently a freshman at Harvard College, probably concentrating in > > mathematics / mathematics and CS. I've done a lot of mathematics in > > high school (math Olympiads) and university (honors linear and > > abstract algebra, real and complex analysis) so far, and I'm > > interested in bringing mathematics and CS together. > > > I've been dealing with sympy for a couple of weeks now and was > > wondering whether it'd be a good idea for GSoC to implement some more > > complicated combinatorial functionality (e.g. graph algorithms, > > generating functions, recurrence relations, operations on sets,...) ? > > > Also, I'm currently working on several little functions for > > computation of the Galois group of quadratic/cubic/quartic > > polynomials; I'll probably send the code in a couple of days. Maybe > > I'll be able to develop some GSoC-like ideas in this direction > > (abstract algebra) as well. > > > -- > > You received this message because you are subscribed to the Google Groups > > "sympy" group. > > To post to this group, send email to sympy@googlegroups.com. > > To unsubscribe from this group, send email to > > sympy+unsubscr...@googlegroups.com. > > For more options, visit this group > > athttp://groups.google.com/group/sympy?hl=en. -- You received this message because you are subscribed to the Google Groups "sympy" group. To post to this group, send email to sympy@googlegroups.com. To unsubscribe from this group, send email to sympy+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
